Predicting probabilities of events using neural networks
You can set up a neural network to predict whether there is an event in a randomly picked interval. I.e. if there is an event in this interval in your trainingsdata you train to output a 1 otherwise you train to output a 0. 

If you use the quadratic loss function the prediction of the NN should approximate the probability of such an event. 

Overfitting can be monitored with by splitting your trainingsdata into train- and testset. 

If you train an RNN by inputting the intervals between events, these intervals should be more similar than the exact event times. Modelling a time series like this also makes more sense. 

Of course these details depend on what exactly the data is representing. And it is also possible that decent predictions are impossible for this dataset.